Certifying Entanglement Dimensionality by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>k</mml:mi> </mml:math> -Reduction Moments

In this paper, we combine the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>k</a:mi> </a:math> -reduction map, the moment method, and randomized measurements into a practical protocol for certifying the entanglement dimensionality. Our approach is based on the observation that a state with entanglement dimensionality at most <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>k</c:mi> </c:math> must stay positive under the action of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>k</e:mi> </e:math> -reduction map. The core of our protocol utilizes the moment method to determine whether the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>k</g:mi> </g:math> -reduced operator, i.e., the operator obtained after applying the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>k</i:mi> </i:math> -reduction map on a quantum state, contains a negative eigenvalue or not. Notably, we propose a systematic method for constructing reduction moment criteria that apply to a broader range of states, including <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>k</k:mi> </k:math> -unfaithful states, compared with fidelity-based methods. The performance of our approach gets better and better with the moment order employed, which is corroborated by extensive numerical simulations. To apply our approach, it suffices to implement a unitary 3-design instead of a 4-design, which is more feasible in practice than the correlation matrix method. In the course of study, we show that the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>k</m:mi> </m:math> -reduction negativity, the absolute sum of the negative eigenvalues of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>k</o:mi> </o:math> -reduced operator, is monotonic under local operations and classical communication for pure states.